#6816703

order	512
nilpotency class	3
isoclinism	isoclinic class of #6814805
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#61294	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1564 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2)(3,10)(4,13)(5,8)(6,9)(7,12)(11,15)(14,16)(17,18)(19,24)(20,23)(21,25)(22,26)(27,30)(28,32)(29,31);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,10)(13,16)(17,18)(19,27)(20,25)(21,23)(22,26)(24,30)(28,31)(29,32);(18,23,26,25)(24,31,32,27);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,10,16)(19,28)(24,32)(27,31)(29,30);(1,5,6,4)(2,8,9,13)(3,11,12,14)(7,16,10,15)(17,19,21,30,22,28,20,29)(18,24,25,27,26,32,23,31)

Polycyclic presentation:

2^9|c^2=gh,d^2=f,e^2=fh,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6723899);